Miracle: Difference between revisions

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'''Miracle temperament''' is a regular temperament discovered by [[George Secor]] in 1974 which has as a generator an interval, called '''secor''', that serves as both [[15/14]] and [[16/15]] semitones.
Miracle is an exceptionally efficient linear temperament which is a member of both the [[Marvel_temperaments|marvel temperaments]] and the [[Gamelismic_clan|gamelismic clan]]. It is quite accurate, with TOP error only 0.63 cents/octave, meaning intervals of the 11-limit tonality diamond are represented with only one or two cents of error. Yet it is also very "low complexity" (efficient), as evidenced by the high density of 11-odd-limit ratios (in bold) in the following table:
Miracle is an exceptionally efficient linear temperament which is a member of both the [[Marvel_temperaments|marvel temperaments]] and the [[Gamelismic_clan|gamelismic clan]]. It is quite accurate, with TOP error only 0.63 cents/octave, meaning intervals of the 11-limit tonality diamond are represented with only one or two cents of error. Yet it is also very "low complexity" (efficient), as evidenced by the high density of 11-odd-limit ratios (in bold) in the following table:


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|-
|-
! | number of <br>secors
! | number of <br>secors
! | cents value* <br>(octave-reduced)
! | JI intervals <br>represented
! | JI intervals <br>represented
|-
|-
| | 0
| | 0
| | 0.00
| | '''1/1'''
| | '''1/1'''
|-
|-
| | 1
| | 1
| | 116.63
| | 16/15, 15/14
| | 16/15, 15/14
|-
|-
| | 2
| | 2
| | 233.27
| | '''8/7'''
| | '''8/7'''
|-
|-
| | 3
| | 3
| | 349.90
| | '''11/9'''
| | '''11/9'''
|-
|-
| | 4
| | 4
| | 466.53
| | 21/16
| | 21/16
|-
|-
| | 5
| | 5
| | 583.16
| | '''7/5'''
| | '''7/5'''
|-
|-
| | 6
| | 6
| | 699.80
| | '''3/2'''
| | '''3/2'''
|-
|-
| | 7
| | 7
| | 816.43
| | '''8/5'''
| | '''8/5'''
|-
|-
| | 8
| | 8
| | 933.06
| | '''12/7'''
| | '''12/7'''
|-
|-
| | 9
| | 9
| | 1049.69
| | '''11/6'''
| | '''11/6'''
|-
|-
| | 10
| | 10
| | 1166.33
| | 88/45, 96/49, 49/25, <br>108/55, 55/28, 63/32
| | 88/45, 96/49, 49/25, <br>108/55, 55/28, 63/32
|-
|-
| | 11
| | 11
| | 82.96
| | 22/21, 21/20
| | 22/21, 21/20
|-
|-
| | 12
| | 12
| | 199.59
| | '''9/8'''
| | '''9/8'''
|-
|-
| | 13
| | 13
| | 316.23
| | '''6/5'''
| | '''6/5'''
|-
|-
| | 14
| | 14
| | 432.86
| | '''9/7'''
| | '''9/7'''
|-
|-
| | 15
| | 15
| | 549.49
| | '''11/8'''
| | '''11/8'''
|-
|-
| | 16
| | 16
| | 666.12
| | 22/15
| | 22/15
|-
|-
| | 17
| | 17
| | 782.76
| | '''11/7'''
| | '''11/7'''
|-
|-
| | 18
| | 18
| | 899.39
| | 42/25, 27/16
| | 42/25, 27/16
|-
|-
| | 19
| | 19
| | 1016.02
| | '''9/5'''
| | '''9/5'''
|-
|-
| | 20
| | 20
| | 1132.65
| | 48/25, 27/14
| | 48/25, 27/14
|-
|-
| | 21
| | 21
| | 49.29
| | 36/35, 33/32
| | 36/35, 33/32
|-
|-
| | 22
| | 22
| | 165.92
| | '''11/10'''
| | '''11/10'''
|-
|-
| | 23
| | 23
| | 282.55
| | 33/28
| | 33/28
|-
|-
| | 24
| | 24
| | 399.19
| | 44/35
| | 44/35
|-
|-
| | 25
| | 25
| | 515.82
| | 27/20
| | 27/20
|-
|-
| | 26
| | 26
| | 632.45
| | 36/25
| | 36/25
|-
|-
| | 27
| | 27
| | 749.08
| | 54/35, 77/50
| | 54/35, 77/50
|-
|-
| | 28
| | 28
| | 865.72
| | 33/20
| | 33/20
|-
|-
| | 29
| | 29
| | 982.35
| | 44/25
| | 44/25
|-
|-
| | 30
| | 30
| | 1098.98
| | 66/35
| | 66/35
|-
|-
| | 31
| | 31
| | 15.62
| | 81/80
| | 81/80
|}
|}
<span style="font-family: Arial,Helvetica,sans-serif;">*in 11-limit POTE tuning</span>
Some temperaments have 11/9 as a "neutral third", meaning it's exactly half of a 3/2 (tempering out 243/242), and other temperaments (in the [[Gamelismic_clan|gamelismic clan]]) have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts. This is in fact the generator of miracle temperament, called a "secor", and it represents both 16/15 and 15/14.
Some temperaments have 11/9 as a "neutral third", meaning it's exactly half of a 3/2 (tempering out 243/242), and other temperaments (in the [[Gamelismic_clan|gamelismic clan]]) have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts. This is in fact the generator of miracle temperament, called a "secor", and it represents both 16/15 and 15/14.


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| | 116.993
| | 116.993
|}
|}
[[Category:miracle]]
[[Category:miracle]]
[[Category:temperament]]
[[Category:temperament]]
[[Category:todo:clarify]]

Revision as of 09:51, 10 May 2019

Miracle temperament is a regular temperament discovered by George Secor in 1974 which has as a generator an interval, called secor, that serves as both 15/14 and 16/15 semitones.

Miracle is an exceptionally efficient linear temperament which is a member of both the marvel temperaments and the gamelismic clan. It is quite accurate, with TOP error only 0.63 cents/octave, meaning intervals of the 11-limit tonality diamond are represented with only one or two cents of error. Yet it is also very "low complexity" (efficient), as evidenced by the high density of 11-odd-limit ratios (in bold) in the following table:

number of
secors 		cents value*
(octave-reduced) 		JI intervals
represented
0 0.00 1/1
1 116.63 16/15, 15/14
2 233.27 8/7
3 349.90 11/9
4 466.53 21/16
5 583.16 7/5
6 699.80 3/2
7 816.43 8/5
8 933.06 12/7
9 1049.69 11/6
10 1166.33 88/45, 96/49, 49/25,
108/55, 55/28, 63/32
11 82.96 22/21, 21/20
12 199.59 9/8
13 316.23 6/5
14 432.86 9/7
15 549.49 11/8
16 666.12 22/15
17 782.76 11/7
18 899.39 42/25, 27/16
19 1016.02 9/5
20 1132.65 48/25, 27/14
21 49.29 36/35, 33/32
22 165.92 11/10
23 282.55 33/28
24 399.19 44/35
25 515.82 27/20
26 632.45 36/25
27 749.08 54/35, 77/50
28 865.72 33/20
29 982.35 44/25
30 1098.98 66/35
31 15.62 81/80

*in 11-limit POTE tuning

Some temperaments have 11/9 as a "neutral third", meaning it's exactly half of a 3/2 (tempering out 243/242), and other temperaments (in the gamelismic clan) have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts. This is in fact the generator of miracle temperament, called a "secor", and it represents both 16/15 and 15/14.

Miracle can also be thought of as a cluster temperament with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 45/44 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63 all tempered together.

Spectrum of Miracle Tunings by Eigenmonzos

Eigenmonzo Secor
8/7 115.587
11/9 115.803
5/4 116.241
7/5 116.502
6/5 116.588 (5 and 7 limit minimax)
11/10 116.591
12/11 116.596
14/11 116.617
7/6 116.641
10/9 116.716 (9 and 11 limit minimax,
Secor's definition of secor)
11/8 116.755
9/7 116.792
4/3 116.993
Retrieved from "https://en.xen.wiki/index.php?title=Miracle&oldid=41682"